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I am currently attempting to use a bivariate normal distribution to identify the most likely range of movement for a blob in computer vision. This itself is not the problem, however; I do not understand how σ plays a role in finding discrete probability contours.

I am not permitted to post images yet since my reputation is too low, but here is a link to a the graph: enter image description here

This is a sample contour plot from Mathematica which displays the Bivariate Probability Density Function with σX = .27 and σY = .54 . μX = 0, μY = 0, and ρ = 0.

I would appreciate it very much if someone could explicate what determines the contour ellipses and how I would go about calculating them for functions of variable σX and σY.

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I am not a Mathematica expert, but it seems as though the values for the level curves were selected such that the level curves represent five even steps from the peak at (0,0) to where the surface "levels out."

Regardless, let's look at the bivariate Gaussian distribution for $\rho = 0$, which implies that $X$ and $Y$ are uncorrelated.

You can write the PDF of this distribution as

$f(x,y) = \frac{1}{2\pi\sigma_x\sigma_y} \exp \left(-\frac{1}{2}\left[\frac{x-\mu_x}{\sigma_x^2}+\frac{y-\mu_y}{\sigma_y^2}-\frac{2(x-\mu_x)(y-\mu_y)}{\sigma_x \sigma_y}\right]\right).$

You can compute this surface in a straightforward manner, and use any contour curve generating algorithm to plot those curves.

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So, these ellipses arent the only contours? Given that the distribution is continuous, there can be any number of contours, correct? –  JT Cho Jul 17 '12 at 15:39
    
Correct. This is a continuous function, which means you can draw as many, or as few, contours as you desire. Contours are not really a natural property of functions; they're something we derive for various purposes, one of them being visualization. In other words, contours are a consequence of properties of a function. If the answer is suitable, please mark it accepted. –  Arkamis Jul 17 '12 at 15:55
    
Yup! Thank you very much for your help :) –  JT Cho Jul 17 '12 at 16:12
    
The contours represent points with the same value for the density. You get them by set the exponent in Ed's formula to a specific value c. You can easily see that defines an ellipse. These contours are convenient because they provide boundaries for probability region. For example you can determine what value c provides a probability region with content p. The area inside the ellipse represents a cumulative bivariate normal probability –  Michael Chernick Jul 18 '12 at 4:51

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